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A036822
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Number of partitions satisfying cn(1,5) = cn(4,5) = 0.
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5
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0, 1, 1, 1, 2, 2, 3, 4, 4, 7, 6, 10, 11, 13, 18, 19, 25, 30, 33, 45, 47, 61, 70, 81, 100, 111, 135, 157, 177, 218, 238, 288, 328, 374, 443, 495, 579, 663, 747, 878, 973, 1134, 1281, 1448, 1670, 1863, 2135, 2414, 2705, 3103
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OFFSET
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1,5
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COMMENTS
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For a given partition cn(i,n) means the number of its parts equal to i modulo n.
Short: (1=4 := 0).
a(n) is the number of partitions with parts congruent to 0, 2 or 3 mod 5. - George Beck, Aug 08 2020
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LINKS
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FORMULA
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G.f.: Product_{k>=1} 1/((1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3))). - Vaclav Kotesovec, Jul 05 2016
a(n) ~ exp(Pi*sqrt(2*n/5)) / (2*sqrt(2*(5+sqrt(5)))*n). - Vaclav Kotesovec, Jul 05 2016
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MAPLE
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c := proc(L, i, n)
local a, p;
a := 0 ;
for p in L do
if modp(p, n) = i then
a := a+1 ;
end if;
end do:
a ;
end proc:
local a , p;
a := 0 ;
for p in combinat[partition](n) do
if c(p, 1, 5) = 0 then
if c(p, 4, 5) = 0 then
a := a+1 ;
end if;
end if;
end do:
a ;
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MATHEMATICA
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nmax = 50; Rest[CoefficientList[Series[Product[1/((1 - x^(5*k)) * (1 - x^(5*k-2)) * (1 - x^(5*k-3))), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2016 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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